arXiv subjects

Leslie Greengard

Publications and source records attributed to Leslie Greengard.

At least 73 records · Page 4Linked to original sources

arXiv2015

Simple and efficient representations for the fundamental solutions of Stokes flow in a half-space

We derive new formulas for the fundamental solutions of slow, viscous flow, governed by the Stokes equations, in a half-space. They are simpler than the classical representations obtained by Blake and collaborators, and can be efficiently implemented using existing fast solvers libraries. We show, for example, that the velocity field induced by a Stokeslet can be annihilated on the boundary (to establish a zero slip condition) using a single reflected Stokeslet combined with a single Papkovich-Neuber potential that involves only a scalar harmonic function. The new representation has a physically intuitive interpretation.

physics.flu-dyn↗

arXiv2014

Inverse Obstacle scattering in two dimensions with multiple frequency data and multiple angles of incidence

We consider the problem of reconstructing the shape of an impenetrable sound-soft obstacle from scattering measurements. The input data is assumed to be the far-field pattern generated when a plane wave impinges on an unknown obstacle from one or more directions and at one or more frequencies. It is well known that this inverse scattering problem is both ill posed and nonlinear. It is common practice to overcome the ill posedness through the use of a penalty method or Tikhonov regularization. Here, we present a more physical regularization, based simply on restricting the unknown boundary to be band-limited in a suitable sense. To overcome the nonlinearity of the problem, we use a variant of Newton's method. When multiple frequency data is available, we supplement Newton's method with the recursive linearization approach due to Chen. During the course of solving the inverse problem, we need to compute the solution to a large number of forward scattering problems. For this, we use high-order accurate integral equation discretizations, coupled with fast direct solvers when the problem is sufficiently large.

math.NA↗

arXiv2014

A fast solver for multi-particle scattering in a layered medium

In this paper, we consider acoustic or electromagnetic scattering in two dimensions from an infinite three-layer medium with thousands of wavelength-size dielectric particles embedded in the middle layer. Such geometries are typical of microstructured composite materials, and the evaluation of the scattered field requires a suitable fast solver for either a single configuration or for a sequence of configurations as part of a design or optimization process. We have developed an algorithm for problems of this type by combining the Sommerfeld integral representation, high order integral equation discretization, the fast multipole method and classical multiple scattering theory. The efficiency of the solver is illustrated with several numerical experiments.

math.NA↗

arXiv2014

A fast semi-direct least squares algorithm for hierarchically block separable matrices

We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data-sparse and can describe many important operators including those derived from asymptotically smooth radial kernels that are not too oscillatory. The algorithm is based on a recursive skeletonization procedure that exposes this sparsity and solves the dense least squares problem as a larger, equality-constrained, sparse one. It relies on a sparse QR factorization coupled with iterative weighted least squares methods. In essence, our scheme consists of a direct component, comprised of matrix compression and factorization, followed by an iterative component to enforce certain equality constraints. At most two iterations are typically required for problems that are not too ill-conditioned. For an $M \times N$ HBS matrix with $M \geq N$ having bounded off-diagonal block rank, the algorithm has optimal $\mathcal{O} (M + N)$ complexity. If the rank increases with the spatial dimension as is common for operators that are singular at the origin, then this becomes $\mathcal{O} (M + N)$ in 1D, $\mathcal{O} (M + N^{3/2})$ in 2D, and $\mathcal{O} (M + N^{2})$ in 3D. We illustrate the performance of the method on both over- and underdetermined systems in a variety of settings, with an emphasis on radial basis function approximation and efficient updating and downdating.

math.NA↗

arXiv2014

The Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scattering

We present a new formulation for the problem of electromagnetic scattering from perfect electric conductors. While our representation for the electric and magnetic fields is based on the standard vector and scalar potentials ${\bf A},ϕ$ in the Lorenz gauge, we establish boundary conditions on the potentials themselves, rather than on the field quantities. This permits the development of a well-conditioned second kind Fredholm integral equation which has no spurious resonances, avoids low frequency breakdown, and is insensitive to the genus of the scatterer. The equations for the vector and scalar potentials are decoupled. That is, the unknown scalar potential defining the scattered field, $ϕ^{Sc}$, is determined entirely by the incident scalar potential $ϕ^{In}$. Likewise, the unknown vector potential defining the scattered field, ${\bf A}^{Sc}$, is determined entirely by the incident vector potential ${\bf A}^{In}$. This decoupled formulation is valid not only in the static limit but for arbitrary $ω\ge 0$.

math-ph↗

arXiv2014

A fast multipole method for the evaluation of elastostatic fields in a half-space with zero normal stress

In this paper, we present a fast multipole method (FMM) for the half-space Green's function in a homogeneous elastic half-space subject to zero normal stress, for which an explicit solution was given by Mindlin (1936). The image structure of this Green's function is unbounded, so that standard outgoing representations are not easily available. We introduce two such representations here, one involving an expansion in plane waves and one involving a modified multipole expansion. Both play a role in the FMM implementation.

math-ph↗

arXiv2014

Visualizing Skin Effects in Conductors with MRI: ${}^7$Li MRI Experiments and Calculations

While experiments on metals have been performed since the early days of NMR (and DNP), the use of bulk metal is normally avoided. Instead, often powders have been used in combination with low fields, so that skin depth effects could be neglected. Another complicating factor of acquiring NMR spectra or MRI images of bulk metal is the strong signal dependence on the orientation between the sample and the radio frequency (RF) coil, leading to non-intuitive image distortions and inaccurate quantification. Such factors are particularly important for NMR and MRI of batteries and other electrochemical devices. Here, we show results from a systematic study combining RF field calculations with experimental MRI of $^7$Li metal to visualize skin depth effects directly and to analyze the RF field orientation effect on MRI of bulk metal. It is shown that a certain degree of selectivity can be achieved for particular faces of the metal, simply based on the orientation of the sample. By combining RF field calculations with bulk magnetic susceptibility calculations accurate NMR spectra can be obtained from first principles. Such analyses will become valuable in many applications involving battery systems, but also metals, in general.

cond-mat.mtrl-sci↗

arXiv2014

Randomized methods for rank-deficient linear systems

We present a simple, accurate method for solving consistent, rank-deficient linear systems, with or without addi- tional rank-completing constraints. Such problems arise in a variety of applications, such as the computation of the eigenvectors of a matrix corresponding to a known eigenvalue. The method is based on elementary linear algebra combined with the ob- servation that if the matrix is rank-k deficient, then a random rank-k perturbation yields a nonsingular matrix with probability 1.

math.NA↗

arXiv2014

Fast and Accurate Evaluation of Nonlocal Coulomb and Dipole-Dipole Interactions via the Nonuniform FFT

We present a fast and accurate algorithm for the evaluation of nonlocal (long-range) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel $U(\bx)$ and a density function $ρ(\bx)=|ψ(\bx)|^2$, for some complex-valued wave function $ψ(\bx)$, permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel $\widehat{U}(\bk)$ has a singularity at the origin $\bk={\bf 0}$ in Fourier (phase) space. Thus, accuracy is lost when using a uniform Cartesian grid in $\bk$ which would otherwise permit the use of the FFT for evaluating the convolution. Here, we make use of a high-order discretization of the Fourier integral, accelerated by the nonuniform fast Fourier transform (NUFFT). By adopting spherical and polar phase-space discretizations in three and two dimensions, respectively, the singularity in $\hat{U}(\bk)$ at the origin is canceled, so that only a modest number of degrees of freedom are required to evaluate the Fourier integral, assuming that the density function $ρ(\bx)$ is smooth and decays sufficiently fast as $\bx \rightarrow \infty$. More precisely, the calculation requires $O(N\log N)$ operations, where $N$ is the total number of discretization points in the computational domain. Numerical examples are presented to demonstrate the performance of the algorithm.

math.NA↗

arXiv2013

Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields

The Debye source representation for solutions to the time harmonic Maxwell equations is extended to bounded domains with finitely many smooth boundary components. A strong uniqueness result is proved for this representation. Natural complex structures are identified on the vector spaces of time-harmonic Maxwell fields. It is shown that in terms of Debye source data, these complex structures are uniformized, that is, represented by a fixed linear map on a fixed vector space, independent of the frequency. This complex structure relates time-harmonic Maxwell fields to constant-k Beltrami fields, i.e. solutions of the equation curl(E) = kE. A family of self-adjoint boundary conditions are defined for the Beltrami operator. This leads to a proof of the existence of zero-flux, constant-k, force-free Beltrami fields for any bounded region in R^3, as well as a constructive method to find them. The family of self-adjoint boundary value problems defines a new spectral invariant for bounded domains in R^3.

math.NA↗

arXiv2013

Efficient sum-of-exponentials approximations for the heat kernel and their applications

In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order $O(\log(\frac{T}δ) (\log(\frac{1}ε)+\log\log(\frac{T}δ)))$ for any $x\in\bbR$ and $δ\leq t \leq T$, where $ε$ is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only $O(\log^2(\frac{T}δ))$ terms for fixed accuracy $ε$. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For $N_S$ points in the spatial discretization and $N_T$ time steps, the cost is $O(N_S N_T \log^2 \frac{T}δ)$ in terms of both memory and CPU time for fixed accuracy $ε$. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.

math.NA↗

arXiv2013

The solution of the scalar wave equation in the exterior of a sphere

We derive new, explicit representations for the solution to the scalar wave equation in the exterior of a sphere, subject to either Dirichlet or Robin boundary conditions. Our formula leads to a stable and high-order numerical scheme that permits the evaluation of the solution at an arbitrary target, without the use of a spatial grid and without numerical dispersion error. In the process, we correct some errors in the analytic literature concerning the asymptotic behavior of the logarithmic derivative of the spherical modified Hankel function. We illustrate the performance of the method with several numerical examples.

math.NA↗

arXiv2013

A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions

We present a new integral equation method for the calculation of two-dimensional scattering from periodic structures involving triple-points (multiple materials meeting at a single point). The combination of a robust and high-order accurate integral representation and a fast direct solver permits the efficient simulation of scattering from fixed structures at multiple angles of incidence. We demonstrate the performance of the scheme with several numerical examples.

math.NA↗

arXiv2013

Norm-preserving discretization of integral equations for elliptic PDEs with internal layers I: the one-dimensional case

We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations (PDEs) in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various L_p norms. We show that high-order accurate Nyström discretization leads to well-conditioned finite dimensional linear systems if and only if the discretization is both norm-preserving in a correctly chosen L_p space and adaptively refined in the internal layer.

math.NA↗

arXiv2013

On the convergence of local expansions of layer potentials

In a recently developed quadrature method (quadrature by expansion or QBX), it was demonstrated that weakly singular or singular layer potentials can be evaluated rapidly and accurately on surface by making use of local expansions about carefully chosen off-surface points. In this paper, we derive estimates for the rate of convergence of these local expansions, providing the analytic foundation for the QBX method. The estimates may also be of mathematical interest, particularly for microlocal or asymptotic analysis in potential theory.

math.NA↗

arXiv2013

Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials

Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior domains. The practical application of these methods, however, requires the accurate evaluation of boundary integrals with singular, weakly singular or nearly singular kernels. Historically, these issues have been handled either by low-order product integration rules (computed semi-analytically), by singularity subtraction/cancellation, by kernel regularization and asymptotic analysis, or by the construction of special purpose "generalized Gaussian quadrature" rules. In this paper, we present a systematic, high-order approach that works for any singularity (including hypersingular kernels), based only on the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Discontinuities in the field across the boundary are permitted. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method. We include accuracy tests for a variety of integral operators in two dimensions on smooth and corner domains.

math.NA↗

arXiv2012

On the efficient representation of the half-space impedance Green's function for the Helmholtz equation

A classical problem in acoustic (and electromagnetic) scattering concerns the evaluation of the Green's function for the Helmholtz equation subject to impedance boundary conditions on a half-space. The two principal approaches used for representing this Green's function are the Sommerfeld integral and the (closely related) method of complex images. The former is extremely efficient when the source is at some distance from the half-space boundary, but involves an unwieldy range of integration as the source gets closer and closer. Complex image-based methods, on the other hand, can be quite efficient when the source is close to the boundary, but they do not easily permit the use of the superposition principle since the selection of complex image locations depends on both the source and the target. We have developed a new, hybrid representation which uses a finite number of real images (dependent only on the source location) coupled with a rapidly converging Sommerfeld-like integral. While our method applies in both two and three dimensions, we restrict the detailed analysis and numerical experiments here to the two-dimensional case.

math.NA↗

arXiv2012

A fast, high-order solver for the Grad-Shafranov equation

We present a new fast solver to calculate fixed-boundary plasma equilibria in toroidally axisymmetric geometries. By combining conformal mapping with Fourier and integral equation methods on the unit disk, we show that high-order accuracy can be achieved for the solution of the equilibrium equation and its first and second derivatives. Smooth arbitrary plasma cross-sections as well as arbitrary pressure and poloidal current profiles are used as initial data for the solver. Equilibria with large Shafranov shifts can be computed without difficulty. Spectral convergence is demonstrated by comparing the numerical solution with a known exact analytic solution. A fusion-relevant example of an equilibrium with a pressure pedestal is also presented.

physics.plasm-ph↗